Comparison, absolute-convergence, and limit-comparison tests for noncompact Henstock–Kurzweil integrals

Let $f,g$ be HK integrable on every compact truncation near the same missing endpoint.

1. If $g\ge 0$, $\mathrm{\mid }f\mathrm{\mid }\le g$ eventually, and the noncompact integral of $g$ converges, then that of $f$ converges.
2. If the noncompact integral of $\mathrm{\mid }f\mathrm{\mid }$ converges, then that of $f$ converges.
3. If $f,g>0$ eventually and $f\mathrm{/}g\to c$ with $0<c<\mathrm{\infty }$, their noncompact integrals converge or diverge together. If $c=0$, convergence for $g$ implies convergence for $f$; if $c=+\mathrm{\infty }$, convergence for $f$ implies convergence for $g$.

The corresponding assertions hold at finite and infinite missing endpoints on either side. At an infinite missing endpoint, the notation $f\mathrm{/}g\to +\mathrm{\infty }$ in claim 3 means explicitly that for every real $M>0$, one has $f\mathrm{/}g>M$ throughout some sufficiently late tail; this clause does not rely on a finite-limit definition.